As in the case of RID=7876 and 9944 (MID=103), the MUSE spatial and spectral resolutions are not always high enough to disentangle the spec-\(z\)  assignment. It can also make it difficult to assign extended \(\mbox{Ly$\alpha $}\)  emission to close HST sources. Following [1]}, we checked if galaxies have close projected neighbors within 06 using the catalog of [2]}. We did not consider MUSE LAEs as neighbors here, because extended \(\mbox{Ly$\alpha $}\)  emission from satellites is one of the candidates for the powering sources of \(\mbox{Ly$\alpha $}\)  haloes. In our sample, 6 sources (RID=4764 , 4838, 5479, 7876, 9944, and 10018) have one HST-detected galaxy within 06. We used the MXDF catalog to check the spec-\(z\)  of the HST neighbors. The HST neighbors of RID=4764 and 10018 are located at different redshifts and do not contaminate the \(\mbox{Ly$\alpha $}\)  NB used in the halo tests (RID=10516 at \(z_{\rm s}=1.42\)  and RID=10046 at \(z_{\rm s}=2.59\) , respectively). Unfortunately, the HST neighbor of RID=5479, RID=5498, does not have a spec-\(z\) , but the photo-\(z\) , \(z_{\rm p}=2.99^{+0.27}_{-0.28}\) , is not close to the spec-\(z\)  of RID=5479, \(z_{\rm s}=4.16\) . Therefore, we regarded these three sources as isolated sources. Meanwhile, the HST neighbor of RID=4838 (RID=6666 at \(z_{\rm s}= 3.06\) ) has a velocity offset of \(\Delta V=-25.4\)  km s\(^{-1}\)  from RID=4838, which indeed contaminates the \(\mbox{Ly$\alpha $}\)  NB used in the halo test (see Figures REF  and REF ). They belong to a group of galaxies within the cosmic web detected with \(\mbox{Ly$\alpha $}\)  emission in [3]}, see Appendix REF  for more details. Since both RID=7876 and 9944 are assigned to counterparts of MID=103 at \(z_{\rm s}=2.09\) , they are interpreted to share \(\mbox{Ly$\alpha $}\)  emission in their \(\mbox{Ly$\alpha $}\)  NBs. In addition, MID=103 has another neighbor within 06, RID=7847 at \(z_{\rm s}=3.00\)  in our sample (\(\Delta V=458\)  km s\(^{-1}\)  from RID=7876 and 9944), which leads to mutual contamination of their \(\mbox{Ly$\alpha $}\)  NBs (see Figures REF  and REF ). Although RID=7847 is located further than 06 from the positions of RID=7876 and 9944, these three sources also belong to a larger structure of cosmic web found in [3]}, as shown in Figure REF . Therefore, RID=4838, 7847, 7876, and 9944 might live in a different kind of environment from the rest of our sample, which may affect the existence of a \(\mbox{Ly$\alpha $}\)  halo and \(\mbox{Ly$\alpha $}\)  halo properties. Additional details of four nonisolated sources are given in Appendix REF . We calculated \(\mbox{Ly$\alpha $}\)  halo fractions for the isolated sources and the nonisolated sources as well as all sources in Section REF .
